Cohomology of vertex algebras

Let V be a vertex algebra and M a V-module. We define the first and second cohomology of V with coefficients in M, and we show that the second cohomology H2(V,M) corresponds bijectively to the set of equivalence classes of square-zero extensions of V by M. In the case that M=V, we show that the second cohomology H2(V,V) corresponds bijectively to the set of equivalence classes of first order deformations of V.